Question

16. In the United States, there has historically been a strong relationship between smoking and education, with well-educated people less likely to smoke. To examine whether this pattern has changed, a sample of 187 high school graduates, a sample of 139 College graduates, and a third sample of 133 graduate school graduates were selected at random from those who had visited a health center for a routine check-up over the course of the past year. Smoking Status Education Nonsmoker Former Moderate Heavy Total High School 56 54 41 36 187 College 37 43 27 32 139 Graduate 53 28 36 16 133 Total 146 125 104 84 459

What type of the chi-square test for this two-way table should be conducted?

What are the degrees of freedom for the chi-square test for this two-way table? What is the p-value and test statistic for the chi-square test for this two-way table? What conclusion can you make?

Answer 1

We Use a chi square test of independence as we need to know if there is any relationship between type of smoker and education level.

The degrees of freedom, df = (r – 1) * (c -1) = (3 - 1) * (4 - 1) = 6

The Expected value data are in the table below. Each Cell = Row total * Column Total/N. N =

Expected
	Non Smoker	Former	Moderate	Heavy	Total
High School	59.48	50.93	42.37	34.22	187
College	44.21	37.85	31.49	25.44	139
Graduate	42.31	36.22	30.14	24.34	133
Total	146	125	104	84	459

The Hypothesis:

H0: There is no relation between Smoking and Education.

Ha: There is a relation between Smoking and Education.

The Test Statistic: The table below gives the calculation of .

#	Observed	Expected	(O-E)	(O-E)2	(O-E)2/E
1	56	59.48	-3.48	12.12071	0.203773
2	37	44.21	-7.21351	52.03469	1.176896
3	53	42.31	10.69499	114.3828	2.703765
4	54	50.93	3.074074	9.449931	0.185562
5	43	37.85	5.145969	26.481	0.699556
6	28	36.22	-8.22004	67.56912	1.865517
7	41	42.37	-1.37037	1.877915	0.044321
8	27	31.49	-4.49455	20.20101	0.641413
9	36	30.14	5.864924	34.39733	1.141438
10	36	34.22	1.777778	3.160494	0.092352
11	32	25.44	6.562092	43.06104	1.69279
12	16	24.34	-8.33987	69.55342	2.857592
Total					13.30498

test = 13.31

The Critical Value: The critical value at = 0.05 (Default level) ,df = 6

critical = 12.59

The p value: The p value at test = 13.31, df = 6; P value = 0.0384

The Decision Rule: If test is > critical, then Reject H0.

If p value is < , Then Reject H0.

The Decision:    Since test (13.31) is > critical (12.59), We Reject H0.

Since p value (0.0384) is < (0.05), We Reject H0 .

The Conclusion: There is sufficient evidence at the 95% significance level to conclude that there is a relationship between smoking and education.